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Cigler’s Motzkin-Column Hankel Formula

Abstract

At every positive shift, each Motzkin-triangle column has Cigler’s specified Hankel-series denominator and a numerator of the exact stated degree.

Theorem 1.1 (The generating function of a Motzkin-column Hankel determinant).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn.result (✓ std3). ∎

Resolves. Problems/cigler-motzkin-column-hankel (proved) by D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumn.result.

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

For every nonnegative integer k and every integer m at least one, let M_{n,k}(t) sum the weights of Motzkin paths from (0,0) to (n,k) never below the axis, with up and down steps of weight one and all horizontal steps of weight t. Put d_m^{(k)}(n,t) = det(M_{m+i+j,k}(t)) for indices i and j from zero to n minus one, with empty determinant one. Define L_0 = 2, L_1 = t and L_{r+2} = t L_{r+1} - L_r, and put e = (-1)^binom(k+1,2). Set A_{k,0}(x,t) = 1 - e x^(k+1) and A_{k,r}(x,t) = 1 - e L_r(t)x^(k+1) + x^(2(k+1)) for positive r. There is an integer polynomial R_m^{(k)}(x,t) of exact x-degree binom(m+1,3) + k(binom(m,1) + binom(m,2) + binom(m,3)) such that the formal series summing d_m^{(k)}(n,t)x^n over all nonnegative n, multiplied by the product of A_{k,(k+1)(m-2j)}(x,t)^(1+j(m-j)) over j from zero through the integer part of m divided by two, equals R_m^{(k)}. This establishes Conjecture 1.3, equation (1.30), of Cigler’s paper. Monic remainder determinants give a mixed alternant with m confluent nodes and k fixed Chebyshev nodes. Reciprocal branches bound its polynomial modes by j(m-j), and the first nonzero negative-index determinant fixes the numerator degree. The identity holds over the integer polynomial ring in t and therefore under every specialization of t. The exact degree is asserted over that ring; specialization can lower it. The specified denominator is a common denominator, without an assertion that numerator and denominator are relatively prime.

References